Both of these terms are readily computed: \(\nabla _\theta \log |\mathcal {F}(\theta )|\)  is known in closed-form for certain families of \({\rm q}\)  such as commonly-used Multivariate Gaussians or can be estimated numerically. \(\nabla _\theta {\rm KL}({\rm q}||{\rm p})\)  (or \(\nabla _\theta {\rm KL}({\rm q}||{\rm p}^*)\) ) is simply the gradient of the objective from classic VI for which numerous estimators exist [1]}, [2]} and that are implemented in a variety of statistics toolboxes such as Stan [3]} or PyMC [4]}. Some common estimators of \(\nabla _\theta {\rm KL}({\rm q}||{\rm p})\)  are stochastic, e.g. by using batches of data [1]}, in which case sampling parameters \(\theta \sim \psi (\theta )\)  can be done using stochastic gradient methods from the MCMC sampling literature [6]}.
